The book Reading, Writing, and Proving: A Closer Look at Mathematics (abbreviated RWP in this review) provides a fresh, interesting, and readable approach to the often-dreaded "Introduction to Proof" class. The book has ambitious goals. It seeks to cultivate in students all three of the skills listed in the title. Often, introduction to proof is construed as just that: an introduction to various proof techniques, with quantifiers, sets, and some epsilons and deltas thrown in for good measure. The authors of this book recognize that there are several closely related skills that need to be worked on together, in order to prepare a student for successful functioning in more advanced mathematics courses.

First, the student has to learn how to read mathematics. This involves not only the obligatory understanding of what quantifiers mean, how implications work, and the endless variations on how to negate "If the stars are green or the white horse is shining, then the world is eleven feet wide", but also the various kinds of techniques that mathematicians use to come to grasp with a new concept or theorem. RWP emphasizes Pólya's four-part framework for problem solving (from his book How to Solve It). Early in the book, the authors spend some time discussing the first step, "Understanding the Problem." It shows the reader, through well-selected illustrations, how to generate specific examples in order to understand a problem, how to draw appropriate pictures, how to work out simple cases, and so forth. RWP includes a convenient summary of "Tips on How to Read Mathematics" in Chapter 10, elaborating on Paul Halmos' famous "Don't just read it; fight it!" slogan. There is also a separate helpful set of "Tips on How to Understand Definitions", in Chapter 5.

Then, the student has to learn to write mathematics. Beautiful examples of proofs containing the right mix of formalization and language are provided throughout the book. There is also a section at the end of Chapter 9 entitled "Tips on Writing Mathematics." Excellent advice and wisdom are provided there, such as a reminder that a variable should only be assigned one meaning in a proof, tips on how to write clearly and concisely, and helpful phrases to help structure a proof (e.g., "Suppose to the contrary").

The student, of course, ultimately has to learn how to generate proofs. The authors take great pains to introduce this in small steps. An all-too-common pitfall for mathematicians who attempt to teach students how to do proofs is to think that it is enough to show a student enough proofs of successively more complicated facts, and eventually there will be an epiphany, and the student will "get it."

A student likened the situation to trying to teach free throws. Of course, it is not enough merely to watch the coach throw free throws while the student watches. However, it is not necessarily enough to stand in front of the line with a basketball and throw the ball at the hoop over and over again either. The coach needs to provide guidance in overall techniques, in fine points, and focus on aspects, on top of having the student practice. RWP puts a great deal of effort into casting the skill of constructing proofs as a problem-solving process. In early discussions of proof techniques, RWP discusses standard simple theorems (such as the irrationality of the square root of 2) by first providing a discussion of how one would execute Pólya's "Understanding the Problem" step, and then the "Devising a Plan" step, and then moving onto writing out the details of the proof by contradiction. The authors even compile "Tips on Putting It All Together", at end of Chapter 11, where they give concrete advice for getting started with proofs.

To learn how to construct proofs, mathematics students not only need to know how to string together implications correctly, but they also have to know how to "debug" proofs, which includes learning how to recognize and fix incorrect and incomplete proofs. The book contains many wonderful exercises of the form "Not a proof", where the students are challenged to find and fix the gaps.

RWP contains more than enough material for a one-semester course, and is designed to give the instructor wide leeway in choosing topics to emphasize. RWP covers the usual material on sets, functions, real analysis type proofs (epsilon-delta), and elementary number theory. It gives a nice mixture of proof techniques, including direct and indirect proofs, cases, mathematical induction, and inequality manipulation. Also, its examples, from the start, include interesting statements about polynomials, which provide a familiar but not stultifying arena in which to state and prove theorems. The later chapters include material on elementary set theory, metric spaces and some point-set considerations, and elementary number theory is taken all the way out to the RSA encryption algorithm. The last chapter provides a wealth of projects, which include both guided and open-ended components, covering mathematically rich ground such as the Cantor set, Pascal's triangle, the irrationality π, and algebraic and transcendental numbers.

In comparison to some other notable books for this class, my impression is that RWP is more conversational and informal than some other standard texts (e.g., How to Prove It, by Dan Velleman, or Chapter Zero, by Carol Schumacher). This book has a rich selection of problems for the student to ponder, in addition to "exercises" that come with hints or complete solutions.

I was charmed by this book and found it quite enticing. I had the opportunity to put the book to the acid test by actually teaching out of the book while reviewing it. My students found the overall style, the abundance of solved exercises, and the wealth of additional historical information and advice in the book exceptionally useful. On the other hand, quite of few of them felt that the book could have been a bit more formal at the beginning (when elementary logic and quantifiers were discussed) and could have provided some more theoretical extensions in the main text, perhaps in appendices like the one on the algebraic properties of the real numbers that actually appears in the text. For example, there are a lot of exercises in the early chapters that deal with elementary number theory facts, yet the formal definition of divisibility is buried in a discussion section following the statement of a theorem.

Using RWP as a textbook, I also uncovered some areas where I feel it could use some improvement. For example, some of the problems could be stated more clearly (e.g., a number of the tautology problems in Chapter 2 confused most of my students with their wording). Furthermore, several of the definitions are not set apart from the text very well (parts of them are bolded, e.g. definition of an equivalence relation or a partition), and some of them feel a bit informal for my taste (e.g., the definition of a set, which is "a collection of objects"). Even though it is almost certainly the case that excess rigor on these points is likely to be confusing, and even mentioning the lack of rigor could potentially be confusing to students, it would seem appropriate to include a disclaimer in a footnote at such points. Such improvements would bring some extra polish to this well-conceived, solidly executed, and very useful textbook.

Velleman, Daniel. How to Prove It: A Structured Approach. Cambridge University Press, 1994.

Maria G. Fung (fungm@wou.edu) is an assistant professor of mathematics at Western Oregon University. Her interests are the mathematical preparation of K-8 teachers and the representation theory of Lie groups.